Research Article · Preprint Mathematics
Finite positivity and correction inertia for the Riemann theta form
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory · Number Theory and Special Functions
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
Abstract
We study the original averaged Riemann theta form and its decomposition into a positive energy and a correction. A verification of the Riemann hypothesis through height T implies strict positivity, uniformly over the averaging parameter, for polynomials of degree at most floor(R/24) whenever 10⁸ ≤ R ≤ 2T − 2. The published verification through 3 × 10¹² therefore gives degree 200,000,000,000. For the endpoint correction, a simultaneous polynomial approximation theorem in a derivative-and-potential norm transfers disjoint inverse-square channels near any fixed finite collection of real zeros into common polynomial subspaces. The contributions of the centers add. Infinitely many simple critical-line zeros then imply, in each parity, negative correction inertia divided by log log N tending to infinity. Opposite channels of dimension Ω(log N) have positive physical form but vanishing physical-to-energy ratio. Together these show that a uniform strict contraction obtained by scalar centering and an arbitrary symmetric correction requires rank ω(log log N). All mixed entries and approximation errors are retained. These are finite-positivity and proof-route results; the physical all-order positivity problem remains open.
Keywords Riemann theta form · polynomial approximation · negative inertia · verified zero height · Meixner–Pollaczek polynomials
Scope of the work
Linear finite verified-height-to-degree positivity; additive finite-center endpoint correction inertia and an unconditional super-loglog consequence; necessary rank obstruction for scalar-centered strict contraction. No negative physical-form polynomial, all-order positivity theorem, RH solution, optimal rate, effective uniform growing-center onset or worldwide priority is claimed.
Reproduction notes
Complete editable TeX sources and portable verification programs are in the publication ZIP. Python 3.10+; python-flint 0.9.0 for two 256-bit Acb xi sign enclosures. Normal and optimized outputs agree. The Platt–Trudgian verification through height 3 × 10¹² is imported, not rerun. Finite constants support analytic proofs; they do not compute large theta inertia matrices.
The manuscript
Read the paper
The complete manuscript, as published in this edition.
If the reader is unavailable on your device, open or download the PDF directly.
Author disclosures
AI tools assisted mathematical development, internal independent checks, literature comparison, computation and manuscript preparation. Internal checks are not external human refereeing or formal proof verification. The author is responsible for the manuscript. No external research funding is declared.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Finite positivity and correction inertia for the Riemann theta form (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/finite-positivity-correction-inertia-riemann-theta-form/v/1
BibTeX
@techreport{luminary_finite_positivity_correction_inertia_riemann_theta_form_v1,
title = {Finite positivity and correction inertia for the Riemann theta form},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/finite-positivity-correction-inertia-riemann-theta-form/v/1}
}Loading discussion…